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Mirrors > Home > MPE Home > Th. List > isosctrlem1 | Structured version Visualization version GIF version |
Description: Lemma for isosctr 26879. (Contributed by Saveliy Skresanov, 30-Dec-2016.) |
Ref | Expression |
---|---|
isosctrlem1 | ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → (ℑ‘(log‘(1 − 𝐴))) ≠ π) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ax-1cn 11211 | . . . . . . 7 ⊢ 1 ∈ ℂ | |
2 | subcl 11505 | . . . . . . 7 ⊢ ((1 ∈ ℂ ∧ 𝐴 ∈ ℂ) → (1 − 𝐴) ∈ ℂ) | |
3 | 1, 2 | mpan 690 | . . . . . 6 ⊢ (𝐴 ∈ ℂ → (1 − 𝐴) ∈ ℂ) |
4 | 3 | adantr 480 | . . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ ¬ 1 = 𝐴) → (1 − 𝐴) ∈ ℂ) |
5 | subeq0 11533 | . . . . . . . . 9 ⊢ ((1 ∈ ℂ ∧ 𝐴 ∈ ℂ) → ((1 − 𝐴) = 0 ↔ 1 = 𝐴)) | |
6 | 5 | notbid 318 | . . . . . . . 8 ⊢ ((1 ∈ ℂ ∧ 𝐴 ∈ ℂ) → (¬ (1 − 𝐴) = 0 ↔ ¬ 1 = 𝐴)) |
7 | 1, 6 | mpan 690 | . . . . . . 7 ⊢ (𝐴 ∈ ℂ → (¬ (1 − 𝐴) = 0 ↔ ¬ 1 = 𝐴)) |
8 | 7 | biimpar 477 | . . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ ¬ 1 = 𝐴) → ¬ (1 − 𝐴) = 0) |
9 | 8 | neqned 2945 | . . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ ¬ 1 = 𝐴) → (1 − 𝐴) ≠ 0) |
10 | 4, 9 | logcld 26627 | . . . 4 ⊢ ((𝐴 ∈ ℂ ∧ ¬ 1 = 𝐴) → (log‘(1 − 𝐴)) ∈ ℂ) |
11 | 10 | imcld 15231 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ ¬ 1 = 𝐴) → (ℑ‘(log‘(1 − 𝐴))) ∈ ℝ) |
12 | 11 | 3adant2 1130 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → (ℑ‘(log‘(1 − 𝐴))) ∈ ℝ) |
13 | 3 | 3ad2ant1 1132 | . . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → (1 − 𝐴) ∈ ℂ) |
14 | 9 | 3adant2 1130 | . . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → (1 − 𝐴) ≠ 0) |
15 | releabs 15357 | . . . . . . . . . 10 ⊢ (𝐴 ∈ ℂ → (ℜ‘𝐴) ≤ (abs‘𝐴)) | |
16 | 15 | adantr 480 | . . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1) → (ℜ‘𝐴) ≤ (abs‘𝐴)) |
17 | breq2 5152 | . . . . . . . . . 10 ⊢ ((abs‘𝐴) = 1 → ((ℜ‘𝐴) ≤ (abs‘𝐴) ↔ (ℜ‘𝐴) ≤ 1)) | |
18 | 17 | adantl 481 | . . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1) → ((ℜ‘𝐴) ≤ (abs‘𝐴) ↔ (ℜ‘𝐴) ≤ 1)) |
19 | 16, 18 | mpbid 232 | . . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1) → (ℜ‘𝐴) ≤ 1) |
20 | recl 15146 | . . . . . . . . . . . 12 ⊢ (𝐴 ∈ ℂ → (ℜ‘𝐴) ∈ ℝ) | |
21 | 20 | recnd 11287 | . . . . . . . . . . 11 ⊢ (𝐴 ∈ ℂ → (ℜ‘𝐴) ∈ ℂ) |
22 | 21 | subidd 11606 | . . . . . . . . . 10 ⊢ (𝐴 ∈ ℂ → ((ℜ‘𝐴) − (ℜ‘𝐴)) = 0) |
23 | 22 | adantr 480 | . . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ≤ 1) → ((ℜ‘𝐴) − (ℜ‘𝐴)) = 0) |
24 | simpl 482 | . . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ≤ 1) → 𝐴 ∈ ℂ) | |
25 | 24 | recld 15230 | . . . . . . . . . 10 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ≤ 1) → (ℜ‘𝐴) ∈ ℝ) |
26 | 1red 11260 | . . . . . . . . . 10 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ≤ 1) → 1 ∈ ℝ) | |
27 | simpr 484 | . . . . . . . . . 10 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ≤ 1) → (ℜ‘𝐴) ≤ 1) | |
28 | 25, 26, 25, 27 | lesub1dd 11877 | . . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ≤ 1) → ((ℜ‘𝐴) − (ℜ‘𝐴)) ≤ (1 − (ℜ‘𝐴))) |
29 | 23, 28 | eqbrtrrd 5172 | . . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ (ℜ‘𝐴) ≤ 1) → 0 ≤ (1 − (ℜ‘𝐴))) |
30 | 19, 29 | syldan 591 | . . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1) → 0 ≤ (1 − (ℜ‘𝐴))) |
31 | resub 15163 | . . . . . . . . . 10 ⊢ ((1 ∈ ℂ ∧ 𝐴 ∈ ℂ) → (ℜ‘(1 − 𝐴)) = ((ℜ‘1) − (ℜ‘𝐴))) | |
32 | re1 15190 | . . . . . . . . . . 11 ⊢ (ℜ‘1) = 1 | |
33 | 32 | oveq1i 7441 | . . . . . . . . . 10 ⊢ ((ℜ‘1) − (ℜ‘𝐴)) = (1 − (ℜ‘𝐴)) |
34 | 31, 33 | eqtrdi 2791 | . . . . . . . . 9 ⊢ ((1 ∈ ℂ ∧ 𝐴 ∈ ℂ) → (ℜ‘(1 − 𝐴)) = (1 − (ℜ‘𝐴))) |
35 | 1, 34 | mpan 690 | . . . . . . . 8 ⊢ (𝐴 ∈ ℂ → (ℜ‘(1 − 𝐴)) = (1 − (ℜ‘𝐴))) |
36 | 35 | adantr 480 | . . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1) → (ℜ‘(1 − 𝐴)) = (1 − (ℜ‘𝐴))) |
37 | 30, 36 | breqtrrd 5176 | . . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1) → 0 ≤ (ℜ‘(1 − 𝐴))) |
38 | 37 | 3adant3 1131 | . . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → 0 ≤ (ℜ‘(1 − 𝐴))) |
39 | neghalfpirx 26523 | . . . . . 6 ⊢ -(π / 2) ∈ ℝ* | |
40 | halfpire 26521 | . . . . . . 7 ⊢ (π / 2) ∈ ℝ | |
41 | 40 | rexri 11317 | . . . . . 6 ⊢ (π / 2) ∈ ℝ* |
42 | argrege0 26668 | . . . . . 6 ⊢ (((1 − 𝐴) ∈ ℂ ∧ (1 − 𝐴) ≠ 0 ∧ 0 ≤ (ℜ‘(1 − 𝐴))) → (ℑ‘(log‘(1 − 𝐴))) ∈ (-(π / 2)[,](π / 2))) | |
43 | iccleub 13439 | . . . . . 6 ⊢ ((-(π / 2) ∈ ℝ* ∧ (π / 2) ∈ ℝ* ∧ (ℑ‘(log‘(1 − 𝐴))) ∈ (-(π / 2)[,](π / 2))) → (ℑ‘(log‘(1 − 𝐴))) ≤ (π / 2)) | |
44 | 39, 41, 42, 43 | mp3an12i 1464 | . . . . 5 ⊢ (((1 − 𝐴) ∈ ℂ ∧ (1 − 𝐴) ≠ 0 ∧ 0 ≤ (ℜ‘(1 − 𝐴))) → (ℑ‘(log‘(1 − 𝐴))) ≤ (π / 2)) |
45 | 13, 14, 38, 44 | syl3anc 1370 | . . . 4 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → (ℑ‘(log‘(1 − 𝐴))) ≤ (π / 2)) |
46 | pirp 26518 | . . . . 5 ⊢ π ∈ ℝ+ | |
47 | rphalflt 13062 | . . . . 5 ⊢ (π ∈ ℝ+ → (π / 2) < π) | |
48 | 46, 47 | ax-mp 5 | . . . 4 ⊢ (π / 2) < π |
49 | 45, 48 | jctir 520 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → ((ℑ‘(log‘(1 − 𝐴))) ≤ (π / 2) ∧ (π / 2) < π)) |
50 | pire 26515 | . . . . . . 7 ⊢ π ∈ ℝ | |
51 | 50 | a1i 11 | . . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ ¬ 1 = 𝐴) → π ∈ ℝ) |
52 | 51 | rehalfcld 12511 | . . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ ¬ 1 = 𝐴) → (π / 2) ∈ ℝ) |
53 | lelttr 11349 | . . . . 5 ⊢ (((ℑ‘(log‘(1 − 𝐴))) ∈ ℝ ∧ (π / 2) ∈ ℝ ∧ π ∈ ℝ) → (((ℑ‘(log‘(1 − 𝐴))) ≤ (π / 2) ∧ (π / 2) < π) → (ℑ‘(log‘(1 − 𝐴))) < π)) | |
54 | 11, 52, 51, 53 | syl3anc 1370 | . . . 4 ⊢ ((𝐴 ∈ ℂ ∧ ¬ 1 = 𝐴) → (((ℑ‘(log‘(1 − 𝐴))) ≤ (π / 2) ∧ (π / 2) < π) → (ℑ‘(log‘(1 − 𝐴))) < π)) |
55 | 54 | 3adant2 1130 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → (((ℑ‘(log‘(1 − 𝐴))) ≤ (π / 2) ∧ (π / 2) < π) → (ℑ‘(log‘(1 − 𝐴))) < π)) |
56 | 49, 55 | mpd 15 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → (ℑ‘(log‘(1 − 𝐴))) < π) |
57 | 12, 56 | ltned 11395 | 1 ⊢ ((𝐴 ∈ ℂ ∧ (abs‘𝐴) = 1 ∧ ¬ 1 = 𝐴) → (ℑ‘(log‘(1 − 𝐴))) ≠ π) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1086 = wceq 1537 ∈ wcel 2106 ≠ wne 2938 class class class wbr 5148 ‘cfv 6563 (class class class)co 7431 ℂcc 11151 ℝcr 11152 0cc0 11153 1c1 11154 ℝ*cxr 11292 < clt 11293 ≤ cle 11294 − cmin 11490 -cneg 11491 / cdiv 11918 2c2 12319 ℝ+crp 13032 [,]cicc 13387 ℜcre 15133 ℑcim 15134 abscabs 15270 πcpi 16099 logclog 26611 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-inf2 9679 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 ax-pre-sup 11231 ax-addf 11232 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-tp 4636 df-op 4638 df-uni 4913 df-int 4952 df-iun 4998 df-iin 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-se 5642 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-isom 6572 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-of 7697 df-om 7888 df-1st 8013 df-2nd 8014 df-supp 8185 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-2o 8506 df-er 8744 df-map 8867 df-pm 8868 df-ixp 8937 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-fsupp 9400 df-fi 9449 df-sup 9480 df-inf 9481 df-oi 9548 df-card 9977 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-div 11919 df-nn 12265 df-2 12327 df-3 12328 df-4 12329 df-5 12330 df-6 12331 df-7 12332 df-8 12333 df-9 12334 df-n0 12525 df-z 12612 df-dec 12732 df-uz 12877 df-q 12989 df-rp 13033 df-xneg 13152 df-xadd 13153 df-xmul 13154 df-ioo 13388 df-ioc 13389 df-ico 13390 df-icc 13391 df-fz 13545 df-fzo 13692 df-fl 13829 df-mod 13907 df-seq 14040 df-exp 14100 df-fac 14310 df-bc 14339 df-hash 14367 df-shft 15103 df-cj 15135 df-re 15136 df-im 15137 df-sqrt 15271 df-abs 15272 df-limsup 15504 df-clim 15521 df-rlim 15522 df-sum 15720 df-ef 16100 df-sin 16102 df-cos 16103 df-pi 16105 df-struct 17181 df-sets 17198 df-slot 17216 df-ndx 17228 df-base 17246 df-ress 17275 df-plusg 17311 df-mulr 17312 df-starv 17313 df-sca 17314 df-vsca 17315 df-ip 17316 df-tset 17317 df-ple 17318 df-ds 17320 df-unif 17321 df-hom 17322 df-cco 17323 df-rest 17469 df-topn 17470 df-0g 17488 df-gsum 17489 df-topgen 17490 df-pt 17491 df-prds 17494 df-xrs 17549 df-qtop 17554 df-imas 17555 df-xps 17557 df-mre 17631 df-mrc 17632 df-acs 17634 df-mgm 18666 df-sgrp 18745 df-mnd 18761 df-submnd 18810 df-mulg 19099 df-cntz 19348 df-cmn 19815 df-psmet 21374 df-xmet 21375 df-met 21376 df-bl 21377 df-mopn 21378 df-fbas 21379 df-fg 21380 df-cnfld 21383 df-top 22916 df-topon 22933 df-topsp 22955 df-bases 22969 df-cld 23043 df-ntr 23044 df-cls 23045 df-nei 23122 df-lp 23160 df-perf 23161 df-cn 23251 df-cnp 23252 df-haus 23339 df-tx 23586 df-hmeo 23779 df-fil 23870 df-fm 23962 df-flim 23963 df-flf 23964 df-xms 24346 df-ms 24347 df-tms 24348 df-cncf 24918 df-limc 25916 df-dv 25917 df-log 26613 |
This theorem is referenced by: isosctrlem2 26877 |
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